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<b>let </b><font color="Maroon">seq</font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/><b>let </b><font color="Maroon">z</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/>



<b>assume </b><a NAME="E1:80"/>
<font color="Maroon">seq</font> is <a href="comseq_3.html#V1" target="_self">summable</a>
 ;<br/>

<a NAME="E2:80"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">seq</font> is <a href="comseq_2.html#V2">convergent</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:80"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> is <a href="comseq_2.html#V2">convergent</a>
 
<b>by </b><i><a class="ref" href="comseq_2.html#T17">COMSEQ_2:17</a></i>;<br/>
<a NAME="E4:80"/><i><font color="Green">E38</font></i>: 
 <a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <font color="Maroon">seq</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span>
 
<b>by </b><i/>;<br/>
 <a href="comseq_3.html#K8" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <font color="Maroon">seq</font></span>)</span> = 
 <a href="comseq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">z</font> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8" target="_self">Sum</a> <font color="Maroon">seq</font></span>)</span>
<b>by </b><i>, <a class="ref" href="comseq_2.html#T18">COMSEQ_2:18</a></i>
;<br/>
<b>hence </b><a NAME="E6:80"/>
( <font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <font color="Maroon">seq</font> is <a href="comseq_3.html#V1" target="_self">summable</a> &amp;  <a href="comseq_3.html#K8" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <font color="Maroon">seq</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8" target="_self">Sum</a> <font color="Maroon">seq</font></span>)</span> )
 <b>by </b><i>, , </i>;<br/>


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